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Conic sections were first studied in 350 B.C. by Menaechmus, who cut a circular conical surface at various angles. Early mathematicians who added to the study of conics include Apollonius, who named them in 220 B.C., and Archimedes, who studied their fascinating properties around 212 B.C. In previous articles in this journal, conic sections have been shown both as an algebraic, or parametric, representation Wonder Embse 1997) and as a geometric, that is, a paper-folding, model (Scher 1996). Both articles offer important insights into the mathematical nature of the conic sections and into teaching methods that can integrate conics into our curriculum. Even though many textbooks discuss conic equations and their graphs, they do not fully develop locus definitions of conic sections.
This article takes the next step in the development of this curricular topic by fully using the power of dynamic-geometry-system (DGS) software to create an environment in which all students can explore and extend the locus definitions of conics. Current versions of popular DGS software, such as Cabri Geometry II, The Geometer's Sketchpad, and TI-92 Geometry, include interactive tools that construct the dynamic loci of points, lines, and other geometric figures. Since the software treats loci as geometric objects, the shape and position of the locus change as the figures in the drawing are manipulated. By dragging and animating points and shapes in the drawings, students can see relationships and connections that they could not see with static representations. The dynamic nature of the locus tool allows students to participate in exploring and investigating the conics. The static pictures presented in this article do not adequately convey the true nature of the dynamic visualizations that students will see when using DGS software.
THE PARABOLA AS A LOCUS OF LINES
What is the locus of points in the plane that are equidistant from a point and a line? Draw a horizontal line d and point F not on the line (fig. 1). The point is called the focus, and the line is called the directrix. Draw a line segment from the focus F to point G anywhere on the directrix, and construct the perpendicular bisector of the segment FG. Construct the locus of the perpendicular bisector as point G moves along the directrix....